MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Corollaire 3.13 in "Champs algebriques" says that the diagonal 1-morphism of stacks $\Delta:\mathcal{X} \to \mathcal{X} \times_S \mathcal{X}$ is representable if and only if the sheaf $\mathcal{Isom}(x,y):\mathrm{Aff}/U \to \mathrm{Ens}$ is represented by an algebraic $U$-space, for every $U \in \mathrm{Aff}/S$ and all $x,y \in \mathcal{X}_U$.

I've seen this shown by claiming that the stack associated to $\mathcal{Isom}(x,y)$ is canonically isomorphic to $U \times_{(x,y),\mathcal{X} \times_S \mathcal{X}, \Delta} \mathcal{X}$. My question is how does one make this identification?

share|cite|improve this question
up vote 2 down vote accepted

This follows immediately from the definitions, in particular from the fiber product of stacks (see loc. cit. (2.2.2)). For $V \in \mathrm{Aff}$ (everything over the base $S$), a $V$-point of $U \times_{X \times X} X$ is a triple $(i,z,\alpha)$, where $i$ is a $V$-point of $U$, $z$ is a $V$-point of $X$ and $\alpha : (i^\* x,i^\* y) \to (i^\*z ,i^\* z)$ is an isomorphism of $V$-points of $X \times X$. This comes down to an isomorphism $\alpha_2^{-1} \circ \alpha_1 : i^\* x \to i^\* y$, i.e. a $V$-point of $\mathrm{Isom}(x,y)$.

share|cite|improve this answer
Thank you! Your answer went to the heart of my difficulties. (I had been having trouble unraveling the definitions.) – dmdmdmdmdmd May 9 '12 at 20:36

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.